# Questions tagged [calculus-of-variations]

Optimization of functionals mostly defined on infinite-dimensional spaces.

1,777 questions
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### Computing Euler Lagrange Equation for a Certain Functional

Let $\Omega\subset \mathbb{R}^n$ be a domain in $\mathbb{R}^n$ with $C^1$ boundary and let $J:\mathscr{H}^{1}_0(\Omega) \to \mathbb{R}$ be given by: $$J(v) = \int_\Omega |v(x)|^p\mathrm{d}x$$ where ...
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### Continuous adjoint of the one-dimensional Laplace equation

Say I have a problem given by the 1D Laplace equation, $$R (T(\alpha), \alpha) = \frac{d^2 T(x)}{dx^2} - \alpha(x) T (x) = 0,$$ with $x \in [0,1]$, Dirichlet boundary conditions on $x=0$ and $x=1$, ...
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### “Quick” proof of the fundamental lemma of calculus of variations

Here's the statement: Let $f \in C([a,b])$ and $H$ be the set $\{h\in C([a,b]):h(a)=h(b)=0\}$. If $\int_a^bf(x)h(x)\,\text{d}x=0$ for all $h\in H$, then $f(x)=0$ for all $x\in [a,b]$. I saw a lot ...
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### Using Calculus of Variations to Find the Maximum Second Moment of Area for Constrained Area

I'm trying to maximize centroidal second moment of area while area is constrained. We know, $$I_{xx} = \int\limits_{D} (y^{2} - y_{c}^{2}) dA$$ Where $$y_{c}$$ is $$\frac{\int\limits_{D} y dA}{A}$$...
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### Arclength integral result does not correspond to intuition

Consider the problem of finding the first variation $\delta_{y_0}J$ for $J(y)=\int_0^1\sqrt{1+y'(x)^2}\,\text{d}x$, given $y_0(x)=ax+b$. I don't know if I'm proceeding correctly and the answer I'm ...
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### Euler–Lagrange equation has no solutions

Show that the Euler–Lagrange equation for the functional: $$I(y) = \int_{0}^{1}y dx$$ subject to y(0) = y(1) = 0 has no solutions. Explain why no extremum for I exists. When forming the E-L ...
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### what does “scalar mean curvature of $\partial E$ with orientation induced by the inner normal to E” mean?

I'm currently reading the paper https://arxiv.org/pdf/1007.3899.pdf and I have a question regarding the mean curverture stuff in it (unfortunately, I don't have any knowledge in differential geometry)-...
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### Existence of a minimizer by means of direct methodes

The following is given: Let $\Omega\subset \mathbb{R}^n$ be a bounded, connected open set with Lipschitz boundary. Let $f\in C^0(\overline{\Omega}\times \mathbb{R}\times \mathbb{R}^n)$, $f=f(x,u,\xi)$,...
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### proving an interpolation inequality in $L^p$ norms

Let $1\le p \le \infty$. Prove that for all $\epsilon >0$ there exists a constant $C>0$ such that \|u'\|_{L^p(\mathbb{R})}\le \epsilon \|u''\|_{L^p(\mathbb{R})}+C\|u\|_{L^p(\mathbb{R})} \;\; \...
Can someone please help me follow and understand the steps of the solution marked with $(*)$ and $(@)$? Why is the dot product used and computed with the unit vector. How does this equal the integral? ...